Local and Multi-Scale Strategies to Mitigate Exponential Concentration in Quantum Kernels

TL;DR

This work addresses the problem of exponential concentration in fidelity-based quantum kernels, where the Gram matrix collapses toward the identity as system size or circuit expressivity grows. It introduces two practical mitigation strategies—local patch-wise kernels and multi-scale kernel mixtures—implemented within Qiskit, and evaluates them across real and synthetic tabular datasets while sweeping the feature dimension $d$ up to 20. By examining diagnostics such as $p50$/$p95$ off-diagonal concentration, $r_{eff}$, and centered alignment with labels, the study shows that locality and scale mixing consistently reshape kernel geometry and yield a richer spectral structure, though improvements in SVM accuracy are dataset-dependent. The findings contribute actionable insights for designing quantum kernel pipelines, suggesting that locality and multiscale mixing can reduce concentration and expand informative kernel directions, with practical implications for scalable quantum-classical learning and hardware-aware deployments.

Abstract

Fidelity-based quantum kernels provide a direct interface between quantum feature maps and classical kernel methods, but they can exhibit exponential concentration: with increasing system size or circuit expressivity, the Gram matrix approaches the identity and suppresses informative similarity structure. We present an empirical study of two mitigation strategies implemented in Qiskit: (i) local (patch-wise) kernels that aggregate subsystem similarities, and (ii) multi-scale kernels that mix local and global similarity across patch granularities. We benchmark baseline, local, and multi-scale kernels under matched preprocessing, splits, and SVM protocols on several tabular datasets, sweeping the feature dimension $d\in\{4,6,\dots,20\}$. We report concentration diagnostics based on off-diagonal kernel statistics, spectral richness via effective rank, and centered alignment with labels. Across datasets, local and multi-scale constructions consistently mitigate concentration and yield richer kernel spectra relative to the global fidelity baseline, while the impact on classification accuracy depends on the dataset and dimension.

Figures (10)

• Figure 1: Off-diagonal concentration (p50) vs. feature dimension $d$ for all datasets. The baseline fidelity kernel concentrates rapidly as $d$ increases (p50 $\to 0$), while the local kernel maintains substantially larger off-diagonal similarities; multi-scale is typically intermediate. All kernels are unit-diagonal normalized.
• Figure 2: Entropy-based effective rank $r_{\mathrm{eff}}(K)$ vs. feature dimension $d$. Local kernels generally preserve a richer spectrum (higher effective rank) than the baseline fidelity kernel; multi-scale typically lies between baseline and local.
• Figure 3: SVM test accuracy vs. feature dimension $d$. For each dataset and $d$, the SVM regularization parameter $C$ is selected by validation accuracy from the fixed grid in Eq. \ref{['eq:svm_C_grid']}, then evaluated once on the test split. Accuracy gains from local and multi-scale kernels are dataset dependent.
• Figure 4: Heatmaps of test-accuracy deltas relative to baseline across datasets (rows) and feature dimensions $d$ (columns).
• Figure 5: Mean test-accuracy delta relative to the baseline kernel, reported per dataset and averaged across feature dimensions $d\in\{4,6,\dots,20\}$. Positive values indicate that the local or multi-scale kernel improves accuracy on average, while negative values indicate a decrease relative to baseline.